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Bring a large pot of salted water to boil over high heat.
In a large skillet over medium-high, combine squash and ¼ cup water. Cover and cook (uncovering to stir occasionally), until tender, 5 to 7 minutes. Uncover and continue to cook, stirring constantly, until liquid evaporates, 2 to 3 minutes.
Add butter, ½ teaspoon salt, and ¼ tsp pepper and stir until butter is melted. Add flour and stir to incorporate. Cook, stirring constantly, until squash mixture thickens and raw flour has had a chance to cook, 2 to 3 minutes. Gradually add milk, stirring constantly. Add garlic powder, cayenne, and thyme and bring to a simmer. Add cheddar and goat cheese, stirring until melted. Remove from heat and let cool slightly.
Add pasta to boiling water and cook 3 minutes short of package instructions. Drain, reserving ½ cup pasta water, and return to pot.
Meanwhile, transfer squash mixture to a blender and puree until smooth (you should have just over 4 cups).
Place the pot with cooked pasta over medium-low heat and stir in sauce until coated. Remove from heat and let cool 5 minutes. (If sauce looks thin, don't worry—it with thicken as it begins to cool.) Add splashes of pasta water to loosen until you've reached your desired consistency.